Calc

# Compound Interest Formula - Explained

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## Compound interest with regular additional contributions - formulae

A lot of people have asked me to include a single formula for compound interest with monthly additions. Believe me when I tell you that it isn't quite as simple as it sounds. In order to work out calculations involving monthly additions, you will need to use two formulae - our original one, listed on the first page of this article, plus the 'future value of a series' one for the monthly additions.

At the request of readers, I've adapted the formula explanation to allow you to calculate periodic additions, not just monthly (added May 2016). These formulae assume that your frequency of compounding is the same as the periodic payment interval (monthly compounding, monthly contributions, etc). If you would like to try a version of the formula that allows you to have a different periodic payment interval to the compounding frequency, please see the 'other formulae' section below.

If the additional deposits are made at the END of the period (end of month, year, etc), here are the two formulae you will need:

Compound interest for principal:

P(1+r/n)(nt)

Future value of a series:

PMT × {[(1 + r/n)(nt) - 1] / (r/n)}

If the additional deposits are made at the BEGINNING of the period (beginning of year, etc), here are the two formulae you will need:

Compound interest for principal:

P(1+r/n)(nt)

Future value of a series:

PMT × {[(1 + r/n)(nt) - 1] / (r/n)} × (1+r/n)

Where:

A = the future value of the investment/loan, including interest
P = the principal investment amount (the initial deposit or loan amount)
PMT = the monthly payment
r = the annual interest rate (decimal)
n = the number of compounds per period (months, years, etc)
t = the number of periods (months, years, etc) the money is invested or borrowed

Let's put this into our example:

An amount of \$5,000 is deposited into a savings account at an annual interest rate of 5%, compounded monthly, with additional deposits of \$100 per month (made at the end of each month). The value of the investment after 10 years can be calculated as follows...

P = 5000. PMT = 100. r = 5/100 = 0.05 (decimal). n = 12. t = 10.

If we plug those figures into the formulae, we get:

Total = [ Compound interest for principal ] + [ Future value of a series ]
Total = [ P(1+r/n)^(nt) ] + [ PMT × (((1 + r/n)^(nt) - 1) / (r/n)) ]
Total = [ 5000 (1 + 0.05 / 12) ^ (12 × 10) ] + [ 100 × (((1 + 0.00416)^(12 × 10) - 1) / (0.00416)) ]
Total = [ 5000 (1.00416) ^ (120) ] + [ 100 × (1.00416)^(120) - 1) / 0.00416) ]
Total = [ 8235.05 ] + [ 100 × (0.647009497690848 / 0.00416) ]
Total = [ 8235.05 ] + [ 15528.23 ]
Total = [ \$23,763.28 ]

So, the investment balance after 10 years is \$23,763.28.

One thing you might notice is that this figure may differ slightly from the figure you get from the compound interest calculator. The reason for this is that the compound interest formula above assumes that the interest calculation occurs before the regular deposit is added on. The calculator, conversely, adds the deposit in first before calculating the interest. Both are legitimate ways of calculating.

For further examples of compound interest formulae for periodic compounding, monthly payments, mortgages and loans, I recommend taking a look at this Wikipedia article.

## Other formulae

A few people have written in to ask for a version of the above formula that takes into consideration the number of periodic payments (both formulae above assume your periodic payments match the frequency of compounding). For example, your money may be compounded quarterly but you're making contributions monthly. In this case, you may wish to try this version of the formula (suggested by Darinth Douglas):

Compound interest for principal:

P(1+r/n)(nt)

Future value of a series:

PMT × p {[(1 + r/n)(nt) - 1] / (r/n)}

(With 'p' being the number of periodic payments in the compounding period)

### Working the formula backwards

If you want to work backwards and find out how much you would need to start with in order to achieve a chosen future value, try the following version of the formula: P = A / ( 1 + r/n ) ^ (nt).

Let's say your goal is to end up with \$10,000 in 5 years, and you can get an 8% interest rate on your savings, compounded monthly. Your calculation would be: P = 10000 / (1 + 0.08/12) ^ (12×5) = \$6712.10. So, you would need to start off with \$6712.10 to achieve your goal.

## Scientific calculator

Need a scientific calculator to help with your calculations? Give this one from web2.0calc.com a try...

If you like this article and have found it useful, I would be very grateful if you would please consider sharing it on social media or on your website/blog. Thank you.

Written by Alastair Hazell